Properties

Degree $2$
Conductor $6025$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.779·2-s − 2.96·3-s − 1.39·4-s + 2.30·6-s − 0.937·7-s + 2.64·8-s + 5.77·9-s + 1.14·11-s + 4.12·12-s + 5.29·13-s + 0.730·14-s + 0.724·16-s + 4.50·17-s − 4.49·18-s − 5.43·19-s + 2.77·21-s − 0.895·22-s − 1.97·23-s − 7.83·24-s − 4.12·26-s − 8.21·27-s + 1.30·28-s − 7.61·29-s − 9.12·31-s − 5.85·32-s − 3.40·33-s − 3.51·34-s + ⋯
L(s)  = 1  − 0.551·2-s − 1.71·3-s − 0.696·4-s + 0.942·6-s − 0.354·7-s + 0.934·8-s + 1.92·9-s + 0.346·11-s + 1.19·12-s + 1.46·13-s + 0.195·14-s + 0.181·16-s + 1.09·17-s − 1.06·18-s − 1.24·19-s + 0.605·21-s − 0.190·22-s − 0.412·23-s − 1.59·24-s − 0.809·26-s − 1.58·27-s + 0.246·28-s − 1.41·29-s − 1.63·31-s − 1.03·32-s − 0.592·33-s − 0.602·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{6025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 0.779T + 2T^{2} \)
3 \( 1 + 2.96T + 3T^{2} \)
7 \( 1 + 0.937T + 7T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
13 \( 1 - 5.29T + 13T^{2} \)
17 \( 1 - 4.50T + 17T^{2} \)
19 \( 1 + 5.43T + 19T^{2} \)
23 \( 1 + 1.97T + 23T^{2} \)
29 \( 1 + 7.61T + 29T^{2} \)
31 \( 1 + 9.12T + 31T^{2} \)
37 \( 1 - 5.51T + 37T^{2} \)
41 \( 1 - 6.55T + 41T^{2} \)
43 \( 1 - 0.588T + 43T^{2} \)
47 \( 1 + 2.18T + 47T^{2} \)
53 \( 1 + 1.78T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 - 7.30T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + 3.52T + 89T^{2} \)
97 \( 1 + 6.10T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.71467819338197560567364000475, −6.93824233190982570775527028751, −6.08090244550342669840003288336, −5.77151301919342209702341296274, −4.99084232571257336519551547435, −4.03843004781108251535626772009, −3.67555452086978177028955097610, −1.77493822026476317732142218906, −0.951052393975359793933545907184, 0, 0.951052393975359793933545907184, 1.77493822026476317732142218906, 3.67555452086978177028955097610, 4.03843004781108251535626772009, 4.99084232571257336519551547435, 5.77151301919342209702341296274, 6.08090244550342669840003288336, 6.93824233190982570775527028751, 7.71467819338197560567364000475

Graph of the $Z$-function along the critical line